To me, the reason it raises hackles is, quite simply, that it's unfair. Alan Bennett was correct.
If you take 2 identical DC in terms of background and intellect; educate one privately, the other in a non-selective state, the chances are heavily weighted in favour of the privately educated one that they will ultimately do better in life, via the commonly accepted denominators of 'better'. They'll be richer; have more choice; live in a nicer area, in a bigger house; will educate their own DC privately and thus the cycle will be perpetuated.
The 'standard B grade' DC will get an A in private; a B in state. Russell Group unis will draw a line according to perceived attainment. (I won't be looking at Oxbridge here as that's a separate issue). The privately educated DC will get the place.
And why shouldn't they?
They 'know the score', no political boats will be rocked; they have been forensically guided in how to maximise marks-per-answer in their exams; they're in small classes, taught by people whose income is dependent on grade-output; they have no disruption or SEN- even 'different ability!- to sideline the teacher. Ideal Russell Group material, I'd've said, were I a RG Admissions Bod.
The other issue not to be overlooked is that if a DC's parents can afford private, it's because they, themselves, have financially succeeded. They have done so because they are a bit 'cleverer' that those around them, and they will pass their 'clever' genes on to their DC who therefore might also be cleverer. I 'get' that this phenomenon can tend to be ignored by the left-wing press.
However, however...
Let's momentarily consider the rise of the proletariat with the advent of The Grammar School. interesting programme. The long and short is that The Establishment needed to educate a lower-ranking officer class in order to intellectually oversee Empire and Industry. These DC were drawn from the great unwashed. At the time, a nod and a wink in the right direction got Alphonse into Eton as he was Of The Manner Born. There was no competition. Then- suddenly- who were these bloody upstarts? Passing O levels, A levels, threatening Alphonse's God Given place at Cambridge? SO Eton, Harrow, Oxbridge, et al, upped their game... or maybe changed their game. Instead of inculcating Classics into Alf, they focused on getting him through these O and A levels. Which, with the level of input to hand, he did. Daddy's old alma mater at Cambs helped of course, but...Equilibrium restored.
Thus, today, although the state educates 93% of DC, RG unis have something like 40% of their intake from the private sector. Allowing for 'clever genes', that's still unequal.
The government are something like 57% privately educated. I need not go there in terms of 'equality'. CEOs and the people who actually are in charge, those with their hand on the financial tiller, are also vastly, if not more so, privately educated.
This isn't because they're over-all 'cleverer', more astute, better than the rest of us, it's because they made a) the contacts and b) learned 'the walk' at private schools. They understand the mutual requirements of keeping each other 'in power' via money and/or influence, therefore, Rich.
The 'state' have done what they thought they could. They got rid of selection (but not in 140-ish schools...), figuring that there's no reason why all DC of all abilities cannot be educated together, though maybe streamed or 'set'. Note this was a Tory policy. But no one DARED take on the power and might of the Private Sector. Too Hard, therefore not addressed. even their charitable status 
. Why? Because those in a position to influence change had their own DC in, at best, private schools; at worst selective grammars. Quite simply. Self-interest writ large.
So, to answer the OP, go, go, go private (good! A good private! Do not be fooled!). You are buying advantage in a world that tacitly accepts this reality. There is nothing 'wrong' with your choice. Yes, it further entrenches the inequalities that hold back and strangles British Society; it may in future, looking back, be recognised as a massively divisive, unfair and ultimately counter-productive force at work in C21 UK; but, if I had the choice, I'd be buying an education that can ride rough-shod over your DC because, when it comes to one's own DC, channelling diane abbott that's all that matters.
And I mean that. Do what you can to advantage your DC over everyone else's. It ain't attractive; it ain't pretty. It's understandable. Just don't tell us you're doing it because of 'the art/drama/sport', thanks.